Discrete Mathematics Past Paper PDF (BTCS-401-18) 2022

Roll No. Total No. of Pages : 03 Total No. of Questions : 18 B.Tech. (Artificial Intelligence & Machine Learning/ Computer Engineering / Computer Science & Engineering / Information Technology/ CSE (Internet of Things and Cyber Security including...

Roll No. Total No. of Pages : 03 Total No. of Questions : 18 B.Tech. (Artificial Intelligence & Machine Learning/ Computer Engineering / Computer Science & Engineering / Information Technology/ CSE (Internet of Things and Cyber Security including Block Chain Technology/Artificial Intelligence & Machine Learning)) (Sem.–4) DISCRETE MATHEMATICS Subject Code : BTCS-401-18 M.Code : 77626 Date of Examination : 02-07-22 Time : 3 Hrs. Max. Marks : 60 INSTRUCTIONS TO CANDIDATES : 1. SECTION-A is COMPULSORY consisting of TEN questions carrying T WO marks each. o m.r c 2. SECTION-B contains FIVE questions carrying FIVE marks each and students have to attempt any FOUR questions. 3. m SECTION-C contains T HREE questions carrying T EN marks each and students have to attempt any T WO questions. o e.r c SECTION-A a p Answer briefly : p e r p 1. a b Give an example of a relation which is reflexive but neither symmetric nor transitive. 2. r p Determine the domain and range of the relation R = {(x, y) : xN. yN and x+y= 10} 3. b How many 8- letter words can be made using the letters of the words "TRIANGLE", if each word is to begin with T and end with E? 4. Define permutation groups. 5. Write down the truth table of (p  q)  r. 6. Is there a simple graph G with six vertices of degree 1, 3, 4, 6, 7? 7. Define a complete binary tree. 8. Give an example of a connected graph that has an Euler circuit but no Hamiltonian circuit. 9. What will be the chromatic number of complete graph with n - vertices? 1 | M-77626 (S2)-65  10. Define equivalent sets. SECTION-B 11. Show that intersection of two partial order relations is a partial order relation. But union of two partial order relations need not be a partial order relation. Give suitable example. 12. The set C* of all non-zero complex numbers form an infinite abelian group under the operation of multiplication of complex numbers. 13. a) How many people must you have to guarantee that at least 5 of them will have birthday on the same month. b) Find the number of positive integers from 1 to 500 which are divisible by at least one of 3, 5 and 7. 14. a) Prove that (p  q)  r = (p  r)  (q  r) o m m b) Prove the validity of the following argument:.r c o p e.r c If a man is bachelor, he is happy. If a man is happy, he dies young. e p a p Therefore bachelors die young. b r p a 15. Show that a graph G with n vertices and (n – 1) edges and no circuit is connected. b r SECTION C 16. Find the shortest path between a and z using Dijkstra’s algorithm for the following graph: b 3 e 2 1 2 5 1 a c z 7 4 2 3 d 4 f 17. a) Prove that every finite integral domain is a field. b) Simplify the Boolean expression f (x, y, z) = (x  y  z)  (x  y  z). And find its conjunctive normal forms. 2 | M-77626 (S2)-65  18. A function f is defined on the set of integers as follows:  1 x 1 x  2  f ( x)   2 x  1 2  x  4 3 x  10 4  x  6  a) Find the domain of the function. b) Find the range of the function. c) Find the value of f (4). d) State whether f is one - one or many one function. o m m.r c o p e.r c p a p e b r p a b r NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any page of Answer Sheet will lead to UMC against the Student. 3 | M-77626 (S2)-65